Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances

An asymmetrical rotating shaft with unequal mass moments of inertia and flexural rigidities in the direction of principal axes is considered. In this system, there are two excitation sources, including a harmonic excitation due to the dynamic imbalances and a parametric excitation due to shaft asymmetry. Nonlinearities are due to the in-extensionality of the shaft and large amplitude. In this study, harmonic and parametric resonances due to the mentioned effects are considered. The influences of inequality of mass moments of inertia and flexural rigidities in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation of steady state response of the rotating asymmetrical shaft are investigated. In addition, the characteristic of stable stationary points and loci of bifurcation points as function of damping coefficient are determined. In order to analyze the resonances of the system the multiple scales method is applied to the complex form of partial differential equations of motion. The achieved results show a good agreement with those of numerical computation.     

具有非轴对称刚度转轴的分岔

研究具有非轴对称刚度转轴的１／２亚谐共振和分岔,首先用Ｈａｍｉｌｔｏｎ原理导出运动微分方程,这是刚度系数周期性变化的参数激励方程,然后用多尺度法求得平均方程分岔响应方程和定常解,最后用奇异性理论分析分岔响应方程和定常解的稳定性,得到了局部分岔集和不同区域的不同分岔响应曲线。     

Bifurcation of a shaft with hysteretic-type internal friction force of material

IntroductionItwasfoundlongtimeagothattheinternalfrictionofmaterialcancauseinstabilityofrotatingshaft.Soitisalwaysoneoftheimportantsubjectsinrotordynamics[1].Earlyinvestigationswerefocusedonthedynamicalstabilityproblemofrotorinfluencedbythelinearinternalfrictionofmaterial,aimingtoobtainthecriterionofstability[2～4 ].Asthedevelopmentofnonlineardynamics,moreandmoreattentionswerepaidtothestudyoftheself_excitedmotionofrotatingshaft,thatisthebifurcation .Thestabilityregionsandbifurcationsofbothanau…     

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

Principal resonance of a nonlinear system with two-frequency parametric and self-excitations

The principal resonance of a single-degree-of-freedom system with two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. Qualitative analyses are employed to study the behaviour of steady state responses, limit cycle responses and 2-torus responses, including their stability and bifurcation. The effects of damping, detuning, and magnitudes of self-excitation and parametric excitations are analyzed. The theoretical analyses are verified by numerical integration results of the governing equation and the modulation equations.     

PRINCIPAL RESPONSE OF VAN DER POL-DUFFING OSCILLATOR UNDER COMBINED DETERMINISTIC AND RANDOM PARAMETRIC EXCITATION

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｒａｎｄｏｍｐａｒａｍｅｔｒｉｃｅｘｃｉｔｅｄｓｙｓｔｅｍｓａｒｅｍｏｒｅｉｍｐｏｒｔａｎｔｔｈａｎｔｈａｔｏｆｒａｎｄｏｍｅｘｔｅｒｎａｌｅｘｃｉｔｅｄｏｎｅｓ,ａｎｄａｒｅｔｈｅｏｒｅｔｉｃａｌｌｙｍｏｒｅｄｉｆｆｉｃｕｌｔｅｓｐｅｃｉａｌｌｙｗｈｅｎｔｈｅｅｘｃｉｔａｔｉｏｎｓａｒｅｎａｒｒｏｗ＿ｂａｎｄｒａｎｄｏｍｐｒｏｃｅｓｓｅｓ[1].Ｉｎｔｈｅｃａｓｅｗｈｅｎｔｈｅｓｙｓｔｅｍｓａｒｅｅｘｃｉｔｅｄｂｙｃｏｍｂｉｎｅｄｄｅｔｅｒｍｉｎｉｓｔｉ…     

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.     

CO-DIMENSION 2 BIFURCATIONS AND CHAOS IN CANTILEVERED PIPE CONVEYING TIME VARYING FLUID WITH THREE-TO-ONE INTERNAL RESONANCES

The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper. The flow velocity is divided into constant and sinusoidai parts. The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes. The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary-differential equations for governing the amplitude of the response. The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters. The codimension 2 derived from the double-zero eigenvaiues is analyzed in detail. The results show that the response amplitude may undergo saddle-node, pitchfork, Hopf, homoclinic loop and period-doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow. When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency, the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.     

VISCO-ELASTIC SYSTEMS UNDER BOTH DETERMINISTIC AND BOUND RANDOM PARAMETRIC EXCITATION

IntroductionThestudyofrandomparametricexcitedsystemsisimportantintheengineeringsystems,andistheoreticallymoredifficultespeciallywhentheexcitationsarenarrow_bandrandomprocessesandparametricexcitedsystems.Inthecasewhenthesystemsareexcitedbycombineddeterministicharmonicandwide_bandprocesses,StratonovitchandRomanovskii[1] ,Dimentbergetal.[2 ] ,andNamachchivaya[3 ] usingthemethodofrandomaveragingandthemethodofKhasminskiiinvestigatedthealmostcertainstabilityofthesystems.AriaratnamandTam[4]discusse…     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

     

Dynamics of axially accelerating beams with multiple supports

This study represents the transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on multiple simple supports. This is one of the examples of a system experiencing Coriolis acceleration component that renders such systems gyroscopic. A small harmonic variation with a constant mean value for the axial velocity is assumed in the problem. The immovable supports introduce nonlinear terms to the equations of motion due to stretching of neutral axis. The method of multiple scales is directly applied to the equations of motion obtained for the general case. Natural frequency equations are presented for multiple support case. Principal parametric resonances and combination resonances are discussed. Solvability conditions are presented for different cases. Stability analysis is conducted for the solutions; approximate stable and unstable regions are identified. Some numerical examples are presented to show the effects of axial speed, number of supports, and their locations.     

复合材料层合板1:1参数共振的分岔研究

针对复合材料对称铺设各向异性矩形层合板的物理模型,在同时考虑了材料、阻尼和几何等非线性因素后,建立了二自由度非线性参数振动系统动力学控制方程,并应用多尺度法求得基本参数共振下的近似解析解,利用数值模拟分析了系统的分岔和混沌运动.指出了伽辽金截断对系统动力学分析的影响,以及系统进入混沌的途径.     

Hopf bifurcation analysis of asymmetrical rotating shafts

The Hopf and double Hopf bifurcations analysis of asymmetrical rotating shafts with stretching nonlinearity are investigated. The shaft is simply supported and is composed of viscoelastic material. The rotary inertia and gyroscopic effect are considered, but, shear deformation is neglected. To consider the viscoelastic behavior of the shaft, the Kelvin–Voigt model is used. Hopf bifurcations occur due to instability caused by internal damping. To analyze the dynamics of the system in the vicinity of Hopf bifurcations, the center manifold theory is utilized. The standard normal forms of Hopf bifurcations for symmetrical and asymmetrical shafts are obtained. It is shown that the symmetrical shafts have double zero eigenvalues in the absence of external damping, but asymmetrical shafts do not have. The asymmetrical shaft in the absence of external damping has a saddle point, therefore the system is unstable. Also, for symmetrical and asymmetrical shafts, in the presence of external damping at the critical speeds, supercritical Hopf bifurcations occur. The amplitude of periodic solution due to supercritical Hopf bifurcations for symmetrical and asymmetrical shafts for the higher modes would be different, due to shaft asymmetry. Consequently, the effect of shaft asymmetry in the higher modes is considerable. Also, the amplitude of periodic solutions for symmetrical shafts with rotary inertia effect is higher than those of without one. In addition, the dynamic behavior of the system in the vicinity of double Hopf bifurcation is investigated. It is seen that in this case depending on the damping and rotational speed, the sink, source, or saddle equilibrium points occur in the system.     

Three-dimensional nonlinear vibrations of composite beams — III. Chordwise excitations

Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Two-mode combination resonances of an in-extensional rotating shaft with large amplitude

In this paper, two-mode combination resonances of a simply supported rotating shaft are investigated. The shaft is modeled as an in-extensional spinning beam with large amplitude. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The equations of motion are derived with the aid of the Hamilton principle and then transformed to the complex form. The method of harmonic balance is applied to obtain analytical solutions. Frequency-response curves are plotted for the combination resonances of the first and the second modes. The effects of eccentricity and external damping are investigated on the steady state response of the rotating shaft. The loci of saddle node bifurcation points are plotted as functions of external damping and eccentricity. The results are validated with numerical simulations.     

Stability and bifurcations analysis of rotating shafts with base excitations

Base excitation in a rotating machinery such as turbo generators, aircraft engines, etc could occur when these systems are subjected to the base movements. This paper investigates the nonlinear behavior of a symmetrical rotating shaft under base excitation when the system is in the vicinity of the main resonance. Dynamic imbalances and harmonic base excitations are the sources of excitation in this system. The equations of motion are derived using the extended Hamilton principle and are mapped into the complex plane. The complex partial differential equation of motion is transformed to the ordinary one utilizing mode shape of the linear system. The method of multiple scales is used to solve the equation of motion. The steady state solutions and their stability are determined, and the effects of damping coefficient, base excitations, and eccentricities of shaft on the stability and bifurcations of the system are investigated. The numerical integration is performed to validate the perturbation results. It is shown that the achieved results from two over-mentioned methods are in accordance with each other.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

Three-dimensional nonlinear vibrations of composite beams — II. flapwise excitations

The nonlinear equations of motion derived in Part I are used to investigate the response of an inextensional, symmetric angle-ply graphite-epoxy beam to a harmonic base-excitation along the flapwise direction. The equations contain bending-twisting couplings and quadratic and cubic nonlinearities due to curvature and inertia. The analysis focuses on the case of primary resonance of the first flexural-torsional (flapwise-torsional) mode when its frequency is approximately one-half the frequency of the first out-of-plane flexural (chordwide) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations to describe the time variation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability and bifurcations of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic solutions of the modulation equations are studied. Chaotic solutions are identified from their frequency spectra, Poincaré sections, and Lyapunov's exponents. The results show that the beam motion may be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Response of nonlinear oscillator under narrow-band random excitation

IntroductionThestudyoftheresponseofnonlinearsystemstonarrow_bandrandomexcitationofconsiderableimportance.Forexample ,theexcitationofsecondarysystemwouldbeanarrow_bandrandomprocessiftheprimarysystemcouldbemodeledasasingle_degree_of_freedomsystemwithlightdampingsubjecttowide_bandexcitation .Inthetheoryofnonlinearrandomvibration ,mostresultsobtainedsofarareattributedtotheresponseofnonlinearoscillatorstowide_bandrandomexcitation .Incomparison ,resultsontheeffectofnarrow_bandexcitationonnonlinearos…     

Parametric resonance of axially moving Timoshenko beams with time-dependent speed

In this paper, parametric resonance of axially moving beams with time-dependent speed is analyzed, based on the Timoshenko model. The Hamilton principle is employed to obtain the governing equation, which is a nonlinear partial-differential equation due to the geometric nonlinearity caused by the finite stretch of the beam. The method of multiple scales is applied to predict the steady-state response. The expression of the amplitude of the steady-state response is derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by using the Lyapunov linearized stability theory. Some numerical examples are presented to demonstrate the effects of speed pulsation and the nonlinearity in the first two principal parametric resonances.